Miami 2011
I. Bandos, mM0 in pp

wave SSP
Multiple M

wave in 11D pp

wave
background and BNM matrix model.
Igor A. Bandos
Based on paper in preparation (2011), Phys.Lett.
B687
(2010),
Phys. Rev. Lett.
105
(2010), Phys, Rev
. D82
(2010).
IKERBASQUE, The Basque Foundation for Science
and
Depto de Física Teórica, Universidad del País Vasco UPV/EHU
, Bilbao, Spain

Introduction

Superembedding approach to M0

brane (M

wave) and multiple M0

brane system (
mM0
= multiple M

wave).

Equations of motion for mM0 in generic 11D SUGRA background

Equations of motion for mM0 in supersymmetric 11D pp

wave
background and
Berenstein

Maldacena

Nastase (
BMN
)
Matrix model.

Conclusion and outlook.
Miami 2011
I. Bandos, mM0 in pp

wave SSP
Introduction
•
Matrix model was proposed by
Banks, Fischler, Shenker and Susskind
in
96

and remains an important tool for studying M

theory.
•
Also the theory is 11 dimensional, original BFSS Lagrangian is just a
dimensional reduction of D=10 SYM down to d=1(low energy mD0). The
symmetry enlargement to D=11 Lorentz symmetry was reveiled by BFSS.
•
However it was not clear how to write the action for Matrix model in 11D
supergravity background.
•
This is why matrix models are known (were guessed) for a few particular
supergravity background, in particular
•
for pp

wave background [
Berenstein

Maldacena

Nastase 2002
] =BMN
Matrix model
•
for matrix Big Bang background [Craps, Sethi, Verlinde 2005]
•
A natural way to resolve this problem was to obtain invariant action, or
covariant equations of motion, for multiple M0

brane system. But it was a
problem to write such action.
•
Purely bosonic mM0 action [
Janssen & Y. Losano
2002]
–
similar to
Myers ‘dielectric’ ‘mDp

brane’ , neither susy nor Lorentz invariance.
•
Superembedding approach to
mM0
system
[
I.B
.
2009

2010] (multiple M

waves or multiple massless superparticle in 11D) =>
SUSY inv. Matrix
model equations in an arbitrary 11D SUGRA.
•
This talk begins the program of develop the application of this result.
•
It is natural to begin by specifying the general equations for pp

wave
background and compear with equations of the BMN matrix model.
Miami 2011
I. Bandos, mM0 in pp

wave SSP
Matrix model eqs. in general 11D
SUGRA background
[I.B. 2010]
Obtained in the frame of superembedding approach
So we need to say what is the superembedding approach
Miami 2011
I. Bandos, mM0 in pp

wave SSP
Bosonic particles and p

branes
.
W
M
Target
space =
spacetime
Worldvolume (worldline)
These worldvolume fields carrying vector indices of D

dimensional spacetime
coordinates are restricted by the
p

brane equations of motion
The embedding of
W
in
Σ
can be
described by coordinate functions
minimal surface equation
(Geodesic eq. for p=0)
Miami 2011
I. Bandos, mM0 in pp

wave SSP
Super

p

branes. Target superspaces and worldvolume
W
Σ
Target D(=11) SUPERspace
Worldvolume (worldline)
The embedding of
W
in
Σ
can be
described by coordinate
functions
These bosonic and fermionic worldvolume fields, carrying indices of 11D superspace
coordinates, are restricted by the super

p

brane equations of motion
Grassmann or
fermionic coordinates
Miami 2011
I. Bandos, mM0 in pp

wave SSP
Superembedding approach,
•
[Bandos,Pasti.Sorokin, Tonin, Volkov 1995, Howe, Sezgin 1996],
•
following the pioneer STV approach to superparticle and
superstring
[Sorokin, Tkach, Volkov MPLA 1989],
•
provides a superfield description of the super

p

brane
dynamics, in terms of embedding of superspaces,
•
namely of embedding of worldvolume superspace
into a target superspace
•
It is thus doubly supersymmetric, and the worldline (world

volume) susy repaces
[STV 89]
the enigmatic local
fermionic kappa

symmetry
[de Azcarraga+Lukierski 82, Siegel
93]
of the standard superparticle and super

p

brane action.
Miami 2011
I. Bandos, mM0 in pp

wave SSP
(M)p

branes in superembedding approach.
I. Target and worldline superspaces.
W
Σ
Target D=11 SUPERspace
Worldvolume (worldline)
SUPERspace
The embedding of
W
in
Σ
can be
described by coordinate functions
These worldvolume superfields carrying indices of 11D superspace coordinates
are restricted by the
superembedding equation
.
Miami 2011
I. Bandos, mM0 in pp

wave SSP
M0

brane (
M

wave
) in superembedding approach.
I. Superembedding equation
W
Σ
Supervielbein of D=11
superspace
Superembedding equation
states that the pull

back of bosonic vielbein
has vanishing fermionic projection:
.
Supervielbein of worldline
superspace
General decomposition of the pull

back of 11D supervielbein
Miami 2011
I. Bandos, mM0 in pp

wave SSP
Superembedding equation
(+ conv. constr.) geometry of worldline superspace
SO(1,1) curvature of
vanishes,
4

form flux of 11D SG= field strength of 3

form
Moving frame vectors
The 4

form flux superfield enters the solution of the 11D superspace SUGRA constraints
[Cremmer & Ferrara 80, Brink & Howe 80] (which results in SUGRA eqs. of motion):
Equations of motion of (single) M0

brane
Miami 2011
I. Bandos, mM0 in pp

wave SSP
Moving frame and spinor moving frame variables
appear in the conventional constraints determining the induced supervielbein so that
Equivalent form of the superembedding eq
.
+ conventional constraints.
M0 equations of motion
Miami 2011
I. Bandos, mM0 in pp

wave SSP
Multiple M0 description by d=1,
=16
SU(N) SYM on
.
•
We describe the multiple M0 by 1d
=16 SYM on
•
The embedding of into the 11D SUGRA superspace
is determined by the superembedding equation
‘center of energy’ motion of the mM0 system is defined by the single M0 eqs
,
Miami 2011
I. Bandos, mM0 in pp

wave SSP
Multiple M0 description by d=1,
=16 SYM on
Basic SYM constraints and superembedding

like equation.
is an su(N) valued 1

form potential on
with the field strength
We impose constraint
A clear candidate for the description of relative
motion of the mM0

constituents!
Bianchi identities DG=0 the
superembedding

like equation
Studying its selfconsistecy conditions, we find the dynamical equations describing
the relative motion of mM0 constituents.
Miami 2011
I. Bandos, mM0 in pp

wave SSP
Equations for the relative motion of multiple M0
in an
arbitrary
supergravity background
follow from the constr.
1d Dirac equation
Gauss constraint
Bosonic equations of motion
Coupling to higher form characteristic for the
Emparan

Myers dielectric brane effect
Miami 2011
I. Bandos, mM0 in pp

wave SSP
mM0 in pp

wave background
Supersymmeric bosonic pp

wave solution of 11D SUGRA
is very well known:
Miami 2011
I. Bandos, mM0 in pp

wave SSP
mM0 in pp

wave background
Miami 2011
I. Bandos, mM0 in pp

wave SSP
•
Thus it looks like we just have to substitute definite pure
bosonic expressions into the general mM0 equations.
•
However, this is not the case, because, for instance
•
and to find some details on the worldline SSP embeddded in
•
and also (in a more general case of, e.g., non

constant flux)
•
as this allows to find
•
To specify our mM0 eqs for some particular SUGRA background
it is necessary to describe this background as a superspace
•
Thus it is not sufficient to know pure bosonic supersymmetric solution
Miami 2011
I. Bandos, mM0 in pp

wave SSP
•
Fortunately the pp

wave superspace is a coset, so that
•
one can write a definite expression for supervielbein etc. (but…)
Miami 2011
I. Bandos, mM0 in pp

wave SSP
•
The worldline (center of energy) superspace is embedded in
this SSP.
•
Its embedding is specified in terms of bosonic and fermionic
coordinate superfields
•
Part of these are Goldstone (super)fields corresponding to
(super)symmetries broken by brane/by center of energy of mM0
•
and part can be identified with coordinates of
•
for instance,
Goldstone fermion superfield
•
Let us begin by the simplest case when the Goldstone
(super)fields describing the center of energy motion are =0, i.e.
by describing a vacuum solution for center of energy SSP
Miami 2011
I. Bandos, mM0 in pp

wave SSP
•
The embedding of the vacuum worldline superspace
•
is characterized by that all Goldstone fields are zero or const.
•
by constant moving frame and spinor moving frame variables
•
by
•
more precisely:
•
One can check that equations of motion and superembedding
equation are satisfied,
•
With which the flux pull

back to
W
is:
Miami 2011
I. Bandos, mM0 in pp

wave SSP
•
But our main interest is in intrinsic geometry of
as the relative motion is described on this superspace.
•
Furthermore, the induced SO(9) and SO(1,1) connection have
only fermionic components, so that
•
What is really important:
Miami 2011
I. Bandos, mM0 in pp

wave SSP
Now we are ready to specify the Matrix
model equations in general 11D SUGRA SSP
1d Dirac equation
Gauss constraint
Bosonic equations of motion
for the case of completely SUSY pp

wave background
Miami 2011
I. Bandos, mM0 in pp

wave SSP
mM0 eqs in pp

wave background
These eqs coincide with the ones which can be obtained by varying the
BMN action up to the fact that they are formulated for traceless matrices.
The trace part of the matrices should describe the center of energy motion.
In our approach it is described separately by the geometry of
To find this, one should go beyond the ground state solution of the superembedding eq,
which we have used above. This is under study now.
Miami 2011
I. Bandos, mM0 in pp

wave SSP
Conclusions and outlook
•
After reviewing of the superembedding approach to mM0
system and the generalization of Matrix model eqs. in an
arbitrary 11D SUGRA background obtained from it
•
we used them to obtain the mM0 equations of motion in the
supersymmetric pp

wave background.
•
The final answer is obtained for a particular susy solution of
the center of energy equations of motion.
•
The equations of the relative motion of mM0 constituents
coincide with the BMN equations, but written for traceless
matrices.
•
To compare the complete set of equations, including the trace
part of the matrices which describe relative motion of the mM0
constituents we need to find general solution of the
superembedding approach equations to M0

brane.
Miami 2011
I. Bandos, mM0 in pp

wave SSP
Some other directions for future study
•
Matrix model equations in AdS(4)xS(7) and AdS(7)xS(4), and their
application, in particular in the frame of AdS/CFT.
Thanks for your attention!
•
Extension of the approach for higher p mDp

and mMp

systems
(mM2

?, mM5

?). Is it consistent to use the same construction (SU(N)
SYM on w/v superspace of a single brane)? And, if not, what is the
critical value of p?
•
To compare the complete set of equations, including the trace
part of the matrices which describe relatove motion of the mM0
constituents we need to find general solution of the
superembedding approach equations to M0

brane.
•
This problem is under investigation now.
Miami 2011
I. Bandos, mM0 in pp

wave SSP
Thank you for
your attention!
Miami 2011
I. Bandos, mM0 in pp

wave SSP
Miami 2011
I. Bandos, mM0 in pp

wave SSP
Appendix A: On BPS equations
for the supersymmetric pure bosonic solutions of mM0 equations
½ BPS equation (16 susy’s preserved)
SUSY preservation by
center of energy motion
SUSY preservation by relative
motion of mM0 constituents
has fuzzy S
² solution modeling M2 brane by mM0 configuration
1/4 BPS equation (8 susy’s) with SO(3) symmetry
and
Nahm equation
Miami 2011
I. Bandos, mM0 in pp

wave SSP
¼ BPS equation (8 susy’s preserved)
which has a(nother) fuzzy 2

sphere

related solution
The famous Nahm equation
appears as an SO(3) inv
with
and
with
is obeyed, in particular, for
½ BPS:
Miami 2011
I. Bandos, mM0 in pp

wave SSP
Miami 2011
I. Bandos, mM0 in pp

wave SSP
Spinor moving frame superfields, entering
are elements of the Spin group valued matrix
`covering’ the moving frame matrix as an SO(1.10) group element
This is to say they are `square roots’ of the light

like moving frame variables, e.g.:
One might wander
whether these spinor moving frame variables come from?
(Auxiliary) moving frame superfields
are elements of the Lorentz group valued matrix
.
This is to say they obey
,
Appendix B: Moving frame and spinor moving frame
Miami 2011
I. Bandos, mM0 in pp

wave SSP
•
Supersymmetric extended objects

super

p

branes

and multiple brane
systems play important role in String/M

theory, AdS/CFT etc.
•
Single p

brane actions
are known for years (84

97)
•
Multiple p

brane actions
(multiple superparticle action for p=0):
•
Multiple Dp

branes (mDp): (very) low energy limit = U(N) SYM (1995)
•
In search for a complete (a more complete) supersymmetric,
diffeomoprhism and Lorenz invariant action =only a particular progress:

purely bosonic mD9 [
Tseytlin
]: non

Abelian BI with symm. trace (no susy);

Myers
[1999] a
`dielectric brane action
’
= no susy, no Loreentz invartiance(!)

Howe, Lindsrom and Wulff
[2005

2007]: the
boundary fermion approach
.
Lorentz and susy inv. action for mDp,
but
on the ‘minus one quantization level’.

I.B.
2009

superembedding approach to
mD0
(proposed for mDp, done for p=0)
•
Multiple Mp

branes (mMp): even more complicated.

purely bosonic mM0= [
Janssen & Y. Losano
2002])

(very) low energy limit = BLG (2007; 3

algebras) ABJM (2008)

nonlinear generalization of (Lorentzian) BLG =[
Iengo & J. Russo
2008]

low energy limit of mM5 = mysterious (2,0) susy d=6 CFT ????

Superembedding approach to
mM0
system
[
I.B
.
2009

2010] (multiple M

waves or multiple massless superparticle in 11D) =>
SUSY inv Matrix model
equations in an arbitrary 11D SUGRA
(
this talk develops its application
).
Introduction
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